Hoi Dick Ng

Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations

To understand the nonlinear dynamical behaviour of a one-dimensional pulsating detonation, results obtained from numerical simulations of the Euler equations with simple one-step Arrhenius kinetics are analysed using basic nonlinear dynamics and chaos theory. To illustrate the transition pattern from a simple harmonic limit-cycle to a more complex irregular oscillation, a bifurcation diagram is constructed from the computational results. Evidence suggests that the route to higher instability modes may follow closely the Feigenbaum scenario of a period-doubling cascade observed in many generic nonlinear systems. Analysis of the one-dimensional pulsating detonation shows that the Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be in reasonable agreement with the universal value of d = 4.669. Using the concept of the largest Lyapunov exponent, the existence of chaos in a one-dimensional unsteady detonation is demonstrated.

Numerical investigation of the instability for one-dimensional Chapman–Jouguet detonations with chain-branching kinetics

The dynamics of one-dimensional Chapman–Jouguet detonations driven by chain-branching kinetics is studied using numerical simulations. The chemical kinetic model is based on a two-step reaction mechanism, consisting of a thermally neutral induction step followed by a main reaction layer, both governed by Arrhenius kinetics. Results are in agreement with previous studies that detonations become unstable when the induction zone dominates over the main reaction layer. To study the nonlinear dynamics, a bifurcation diagram is constructed from the computational results. Similar to previous results obtained with a single-step Arrhenius rate law, it is shown that the route to higher instability follows the Feigenbaum route of a period-doubling cascade. The corresponding Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be close to the universal value of 4.669. The present parametric analysis determines quantitatively the relevant non-dimensional parameter χ, defined as the activation energy for the induction process ϵ I multiplied by the ratio of the induction length Δ I to the reaction length Δ R . The reaction length Δ R is estimated by the inverse of the maximum thermicity (1/ max) multiplied by the Chapman–Jouguet particle velocity u CJ . An attempt is made to provide a physical explanation of this stability parameter from the coherence concept. A series of computations is carried out to obtain the neutral stability curve for one-dimensional detonation waves over a wide range of chemical parameters for the model. These results are compared with those obtained from numerical simulations using detailed chemistry for some common gaseous combustible mixtures.

The effect of argon dilution on the stability of acetylene/oxygen detonations

Experimental observations indicate that dilutions with large amounts of argon lead to stable detonations having a regular cellular structure with only weak transverse waves. In the present study, the stabilizing effect of argon dilution in acetylene/oxygen detonations is investigated numerically by detailed numerical simulations of one-dimensional time-dependent pulsating detonations with a realistic seven-step chemistry model. The results show that heavy argon dilution in the mixture leads to single-frequency small-amplitude regular oscillations of the shock front pressure. As the dilution is decreased, the detonations become unstable, characterized by larger amplitude oscillations. The stabilizing role of argon is further investigated by analyzing the reaction zone structure of the steady Zeldovich-Von Neumann-Doring detonation with varying degrees of argon dilution. For the same characteristic induction lengths, the dilution with argon leads to lower temperatures in the reaction zone and slower exothermic reaction rates, thus rendering the reaction zone structure less temperature sensitive and more stable to hydrodynamic fluctuations. The present unsteady numerical simulations also indicate that with argon dilution less than approximately70%, a one-dimensional time-dependent detonation cannot self-propagate. Below this limit, due to the low-velocity excursions of the pulsating leading shock, the reactions are quenched and detonation failure occurs. This fundamental limit reveals that a one-dimensional shock-induced ignition mechanism in an unstable detonation is insufficient to account for the ignition and propagation mechanism in multidimensional detonations. These observations are consistent with recent experiments performed in porous wall tubes where, as the transverse waves were eliminated, the detonation could not self-sustain in the one-dimensional limit. The experimental stability limit, at which the transverse waves begin to play the dominant role, also corresponding to the loss of regularity in the cellular pattern, agrees very well with the stability limit determined numerically in the present study.

Direct initiation of detonation with a multi-step reaction scheme

The problem of direct initiation of detonation, where a powerful ignition source drives a blast wave into a gaseous combustible mixture to generate a Chapman–Jouguet (CJ) detonation, is investigated numerically by using a three-step chain-branching chemical kinetic model. The reaction scheme consists sequentially of a chain-initiation and a chain-branching step, followed by a temperature-independent chain termination. The three regimes of direct initiation i.e. subcritical, critical and supercritical, are numerically simulated for planar, cylindrical and spherical geometries using the present three-step chemical kinetic model. It is shown that the use of a more detailed reaction mechanism allows a well-defined value for the critical initiation energy to be determined. The numerical results demonstrate that detonation instability plays an important role in the initiation process. The effect of curvature for cylindrical and spherical geometries has been found to enhance the instability of the detonation wave and thus influence the initiation process. The results of these simulations are also used to provide further verification of some existing theories of direct initiation of detonation. It appears that these theories are satisfactory only for stable detonation waves and start to break down for highly unstable detonations because they are based on simple blast wave theory and do not include a parameter to model the detonation instability. This study suggests that a stability parameter, such as the ratio between the induction and reaction length, should be considered and a more complex chemistry should be included in future development of a more rigorous theory for direct initiation of detonation.

Numerical study of oblique detonation wave initiation in a stoichiometric hydrogen-air mixture

Two-dimensional, oblique detonations induced by a wedge are simulated using the reactive Euler equations with a detailed chemical reaction model. The focus of this study is on the oblique shock-to-detonation transition in a stoichiometric hydrogen-air mixture. A combustible, gas mixture at low pressure and high temperature, corresponding to the realistic, inflow conditions applied in oblique detonation wave engines, is presented in this study. At practical flight conditions, the present numerical results illustrate that oblique detonation initiation is achieved through a smooth transition from a curved shock, which differs from the abrupt transition depicted in the previous studies. The formation mechanism of this smooth transition is discussed and a quantitative analysis is carried out by defining a characteristic length for the initiation process. The dependence of the initiation length on different parameters including the wedge angle, flight Mach number, and inflow Mach number is discussed. Despite the hypothetical nature of the simulation configuration, the present numerical study uses parameters we deem relevant to practical conditions and provides important observations for which future investigations can benefit from in reaching toward a rigorous theory of the formation and self-sustenance of oblique detonation waves.

Visualization of an imploding circular wave front and the formation of a central vertical jet

Graphical abstract

Numerical simulation of detonation structures using a thermodynamically consistent and fully conservative reactive flow model for multi-component computations

This paper presents a simplified reactive multi-gas model for the numerical simulation of detonation waves. The mathematical model is formulated based on a thermodynamically consistent and fully conservative formulation, and is extended to model reactive flow by considering the reactant and product gases as two constituents of the system and modelling the conversion between these by a simple one-step reaction mechanism. This simplified model allows simulations using more appropriate chemico-thermodynamic properties of the combustible mixture and yields close Chapman–Jouguet detonation parameters from detailed chemistry. The governing equations are approximated using a high-resolution finite volume centred scheme in an adaptive mesh refinement code, permitting high-resolution simulations to be performed at flow regions of interest. The algorithm is tested and validated by comparing results to predictions of the one-dimensional linear stability analysis of the steady detonation and through the study of the evolution of two-dimensional cellular detonation waves in gaseous hydrogen-based mixtures.

Initiation structure of oblique detonation waves behind conical shocks

The understanding of oblique detonation dynamics has both inherent basic research value for high-speed compressible reacting flow and propulsion application in hypersonic aerospace systems. In this study, the oblique detonation structures formed by semi-infinite cones are investigated numerically by solving the unsteady, two-dimensional axisymmetric Euler equations with a one-step irreversible Arrhenius reaction model. The present simulation results show that a novel wave structure, featured by two distinct points where there is close-coupling between the shock and combustion front, is depicted when either the cone angle or incident Mach number is reduced. This structure is analyzed by examining the variation of the reaction length scale and comparing the flow field with that of planar, wedge-induced oblique detonations. Further simulations are performed to study the effects of chemical length scale and activation energy, which are both found to influence the formation of this novel structure. The initiat...

Applying nonlinear dynamic theory to one-dimensional pulsating detonations

The dynamical behaviour of one-dimensional pulsating detonations was investigated in detail, with the aid of nonlinear theory tools such as phase plots and correlation dimension. The period-doubling cascade, as routes to deterministic chaos, is depicted through the transformations of the shapes of the attractors. Using a correlation dimension method, the dimension of the attractors is determined and we show that the chaos within a one-dimensional pulsating detonation is deterministic.

High resolution GPU-based flow simulation of the gaseous methane-oxygen detonation structure

Graphical abstract